Average Error: 0.5 → 0.5
Time: 31.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}\right)
double f(double v) {
        double r251380 = 1.0;
        double r251381 = 5.0;
        double r251382 = v;
        double r251383 = r251382 * r251382;
        double r251384 = r251381 * r251383;
        double r251385 = r251380 - r251384;
        double r251386 = r251383 - r251380;
        double r251387 = r251385 / r251386;
        double r251388 = acos(r251387);
        return r251388;
}
↓
double f(double v) {
        double r251389 = 1.0;
        double r251390 = 5.0;
        double r251391 = v;
        double r251392 = r251391 * r251391;
        double r251393 = r251390 * r251392;
        double r251394 = exp(r251393);
        double r251395 = log(r251394);
        double r251396 = r251389 - r251395;
        double r251397 = r251392 - r251389;
        double r251398 = r251396 / r251397;
        double r251399 = acos(r251398);
        double r251400 = log1p(r251399);
        double r251401 = sqrt(r251400);
        double r251402 = r251389 - r251393;
        double r251403 = r251402 / r251397;
        double r251404 = acos(r251403);
        double r251405 = log1p(r251404);
        double r251406 = sqrt(r251405);
        double r251407 = r251401 * r251406;
        double r251408 = expm1(r251407);
        return r251408;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.5

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}\right)\]
  6. Using strategy rm

  7. Applied add-log-exp0.5

    \[\leadsto \mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \color{blue}{\log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}\right)\]

  8. Final simplification0.5

    \[\leadsto \mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))